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Let $X$ be a topological space which has the homotopy-type of a CW-complex. As well-known, a CW-complex is locally contractible.

Question: Is $X$ locally contractible? If not, can some one give a counterexample?

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    $\begingroup$ Hatcher's AT book contains an example of a space which is contractible but not locally contractible, so this is in particular a counterexample to your question. I think it was something like $([0,1]\cap \mathbb{Q}) \times [0,1] \cup [0,1]\times\{0\}$. $\endgroup$
    – Achim Krause
    Commented Jun 21 at 9:40

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The best you can do is that if $X$ is homotopy equivalent to a locally contractible space, then $X$ is semilocally contractible: every $x\in X$ has a neighborhood $U$ whose inclusion $U\subseteq X$ is nullhomotopic.

Any contractible space is semilocally contractible (even of CW homotopy type), but may fail to be locally contractible. The comb space is an example.

More generally, one can find counterexamples by taking the cone over any non-locally contractible space.

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